{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "47c70580",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: MASS\n",
      "\n",
      "Loading required package: Matrix\n",
      "\n",
      "Loading required package: mvtnorm\n",
      "\n",
      "Loading required package: sandwich\n",
      "\n",
      "mediation: Causal Mediation Analysis\n",
      "Version: 4.5.0\n",
      "\n",
      "\n",
      "Loading required package: nlme\n",
      "\n",
      "Warning message:\n",
      "“package ‘lavaan’ was built under R version 4.2.3”\n",
      "This is lavaan 0.6-17\n",
      "lavaan is FREE software! Please report any bugs.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "library(mediation)\n",
    "library(multilevel)\n",
    "library(lavaan)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1353a5cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 6 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>X</th><th scope=col>M</th><th scope=col>Y</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>6</td><td>5</td><td>6</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>7</td><td>5</td><td>5</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>7</td><td>7</td><td>4</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>8</td><td>4</td><td>8</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>4</td><td>3</td><td>5</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>4</td><td>4</td><td>7</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       "  & X & M & Y\\\\\n",
       "  & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 & 6 & 5 & 6\\\\\n",
       "\t2 & 7 & 5 & 5\\\\\n",
       "\t3 & 7 & 7 & 4\\\\\n",
       "\t4 & 8 & 4 & 8\\\\\n",
       "\t5 & 4 & 3 & 5\\\\\n",
       "\t6 & 4 & 4 & 7\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 3\n",
       "\n",
       "| <!--/--> | X &lt;int&gt; | M &lt;int&gt; | Y &lt;int&gt; |\n",
       "|---|---|---|---|\n",
       "| 1 | 6 | 5 | 6 |\n",
       "| 2 | 7 | 5 | 5 |\n",
       "| 3 | 7 | 7 | 4 |\n",
       "| 4 | 8 | 4 | 8 |\n",
       "| 5 | 4 | 3 | 5 |\n",
       "| 6 | 4 | 4 | 7 |\n",
       "\n"
      ],
      "text/plain": [
       "  X M Y\n",
       "1 6 5 6\n",
       "2 7 5 5\n",
       "3 7 7 4\n",
       "4 8 4 8\n",
       "5 4 3 5\n",
       "6 4 4 7"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Source: https://library.virginia.edu/data/articles/introduction-to-mediation-analysis\n",
    "# X: grades; Y: happiness; M: self-esteem\n",
    "myData = read.csv('http://static.lib.virginia.edu/statlab/materials/data/mediationData.csv')\n",
    "head(myData)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c34146d2",
   "metadata": {},
   "source": [
    "## Baron & Kenny’s 4-step method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8fd54b14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Y ~ X, data = myData)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-5.0262 -1.2340 -0.3282  1.5583  5.1622 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   2.8572     0.6932   4.122 7.88e-05 ***\n",
       "X             0.3961     0.1112   3.564 0.000567 ***\n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Residual standard error: 1.929 on 98 degrees of freedom\n",
       "Multiple R-squared:  0.1147,\tAdjusted R-squared:  0.1057 \n",
       "F-statistic:  12.7 on 1 and 98 DF,  p-value: 0.0005671\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Step 1: Total effect = Path c = 0.3961***\n",
    "model.0 = lm(Y ~ X, data=myData)\n",
    "summary(model.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "88235b89",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = M ~ X, data = myData)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.3046 -0.8656  0.1344  1.1344  4.6954 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  1.49952    0.58920   2.545   0.0125 *  \n",
       "X            0.56102    0.09448   5.938 4.39e-08 ***\n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Residual standard error: 1.639 on 98 degrees of freedom\n",
       "Multiple R-squared:  0.2646,\tAdjusted R-squared:  0.2571 \n",
       "F-statistic: 35.26 on 1 and 98 DF,  p-value: 4.391e-08\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Step 2: Path a = 0.56102***\n",
    "model.XM = lm(M ~ X, data=myData)\n",
    "summary(model.XM)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "48ed6977",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Y ~ X + M, data = myData)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-3.7631 -1.2393  0.0308  1.0832  4.0055 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   1.9043     0.6055   3.145   0.0022 ** \n",
       "X             0.0396     0.1096   0.361   0.7187    \n",
       "M             0.6355     0.1005   6.321 7.92e-09 ***\n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Residual standard error: 1.631 on 97 degrees of freedom\n",
       "Multiple R-squared:  0.373,\tAdjusted R-squared:  0.3601 \n",
       "F-statistic: 28.85 on 2 and 97 DF,  p-value: 1.471e-10\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Step 3 & 4: Path b + Path c'\n",
    "#             Path b = 0.6355***\n",
    "#             Path c'= 0.0396\n",
    "model.MY.XY = lm(Y ~ X + M, data=myData)\n",
    "summary(model.MY.XY)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49cfe740",
   "metadata": {},
   "source": [
    "Interpretation:\n",
    "- The total effect model shows a significant positive relationship between grades(X) and happiness(Y).\n",
    "    - Path c = 0.3961***\n",
    "- The path a model shows grades(X) positively correlate with self-esteem(M) and the relationship is statistically significant.\n",
    "    - Path a = 0.56102***\n",
    "- Self-esteem(M) positively predicts happiness(Y) when controlling grades(X). At the same time, the relationship between grades(X) and happiness(Y) is no longer significant.\n",
    "    - Path b = 0.6355***\n",
    "    - Path c'= 0.0396\n",
    "- This suggests that self-esteem(M) completely mediates the relationship between grades(X) and happiness(Y).\n",
    "    - However, this method cannot directly evaluate the significance of the indirect effect (i.e. ab≠0) - we need a formal test for a statistical verification."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f03de5c",
   "metadata": {},
   "source": [
    "## Sobel Test\n",
    "\n",
    "Determine the significance of the indirect effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d01577b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'indirect effect: 0.356522196728078'"
      ],
      "text/latex": [
       "'indirect effect: 0.356522196728078'"
      ],
      "text/markdown": [
       "'indirect effect: 0.356522196728078'"
      ],
      "text/plain": [
       "[1] \"indirect effect: 0.356522196728078\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "'z-statistic: 4.32789111803173'"
      ],
      "text/latex": [
       "'z-statistic: 4.32789111803173'"
      ],
      "text/markdown": [
       "'z-statistic: 4.32789111803173'"
      ],
      "text/plain": [
       "[1] \"z-statistic: 4.32789111803173\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "'p-value: 1.5054385481994e-05'"
      ],
      "text/latex": [
       "'p-value: 1.5054385481994e-05'"
      ],
      "text/markdown": [
       "'p-value: 1.5054385481994e-05'"
      ],
      "text/plain": [
       "[1] \"p-value: 1.5054385481994e-05\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 手动计算: https://www.spss-tutorials.com/sobel-test-what-is-it/\n",
    "alpha = summary(model.XM)$coefficients[\"X\",\"Estimate\"]\n",
    "se_alpha = summary(model.XM)$coefficients[\"X\",\"Std. Error\"]\n",
    "beta = summary(model.MY.XY)$coefficients[\"M\",\"Estimate\"]\n",
    "se_beta = summary(model.MY.XY)$coefficients[\"M\",\"Std. Error\"]\n",
    "\n",
    "z.statistic = (alpha * beta)/(sqrt(alpha*alpha*se_beta*se_beta + beta*beta*se_alpha*se_alpha))\n",
    "p.value = 2*(1-pnorm(z.statistic))  # two-tailed\n",
    "\n",
    "sprintf(\"indirect effect: %s\", alpha*beta)\n",
    "sprintf(\"z-statistic: %s\", z.statistic)\n",
    "sprintf(\"p-value: %s\", p.value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0c2f0ef6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$`Mod1: Y~X`</dt>\n",
       "\t\t<dd><table class=\"dataframe\">\n",
       "<caption>A matrix: 2 × 4 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>2.8572046</td><td>0.6932130</td><td>4.121684</td><td>7.875771e-05</td></tr>\n",
       "\t<tr><th scope=row>pred</th><td>0.3961261</td><td>0.1111598</td><td>3.563574</td><td>5.671128e-04</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$`Mod2: Y~X+M`</dt>\n",
       "\t\t<dd><table class=\"dataframe\">\n",
       "<caption>A matrix: 3 × 4 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>1.90426882</td><td>0.6054585</td><td>3.145168</td><td>2.203831e-03</td></tr>\n",
       "\t<tr><th scope=row>pred</th><td>0.03960392</td><td>0.1096484</td><td>0.361190</td><td>7.187429e-01</td></tr>\n",
       "\t<tr><th scope=row>med</th><td>0.63549459</td><td>0.1005339</td><td>6.321194</td><td>7.922642e-09</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$`Mod3: M~X`</dt>\n",
       "\t\t<dd><table class=\"dataframe\">\n",
       "<caption>A matrix: 2 × 4 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>1.4995183</td><td>0.58919773</td><td>2.545017</td><td>1.248749e-02</td></tr>\n",
       "\t<tr><th scope=row>pred</th><td>0.5610153</td><td>0.09448048</td><td>5.937897</td><td>4.391362e-08</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$Indirect.Effect</dt>\n",
       "\t\t<dd>0.356522196728078</dd>\n",
       "\t<dt>$SE</dt>\n",
       "\t\t<dd>0.0823778110411936</dd>\n",
       "\t<dt>$z.value</dt>\n",
       "\t\t<dd>4.32789111803173</dd>\n",
       "\t<dt>$N</dt>\n",
       "\t\t<dd>100</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$`Mod1: Y\\textasciitilde{}X`] A matrix: 2 × 4 of type dbl\n",
       "\\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 2.8572046 & 0.6932130 & 4.121684 & 7.875771e-05\\\\\n",
       "\tpred & 0.3961261 & 0.1111598 & 3.563574 & 5.671128e-04\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$`Mod2: Y\\textasciitilde{}X+M`] A matrix: 3 × 4 of type dbl\n",
       "\\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 1.90426882 & 0.6054585 & 3.145168 & 2.203831e-03\\\\\n",
       "\tpred & 0.03960392 & 0.1096484 & 0.361190 & 7.187429e-01\\\\\n",
       "\tmed & 0.63549459 & 0.1005339 & 6.321194 & 7.922642e-09\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$`Mod3: M\\textasciitilde{}X`] A matrix: 2 × 4 of type dbl\n",
       "\\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 1.4995183 & 0.58919773 & 2.545017 & 1.248749e-02\\\\\n",
       "\tpred & 0.5610153 & 0.09448048 & 5.937897 & 4.391362e-08\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$Indirect.Effect] 0.356522196728078\n",
       "\\item[\\$SE] 0.0823778110411936\n",
       "\\item[\\$z.value] 4.32789111803173\n",
       "\\item[\\$N] 100\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$`Mod1: Y~X`\n",
       ":   \n",
       "A matrix: 2 × 4 of type dbl\n",
       "\n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(&gt;|t|) |\n",
       "|---|---|---|---|---|\n",
       "| (Intercept) | 2.8572046 | 0.6932130 | 4.121684 | 7.875771e-05 |\n",
       "| pred | 0.3961261 | 0.1111598 | 3.563574 | 5.671128e-04 |\n",
       "\n",
       "\n",
       "$`Mod2: Y~X+M`\n",
       ":   \n",
       "A matrix: 3 × 4 of type dbl\n",
       "\n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(&gt;|t|) |\n",
       "|---|---|---|---|---|\n",
       "| (Intercept) | 1.90426882 | 0.6054585 | 3.145168 | 2.203831e-03 |\n",
       "| pred | 0.03960392 | 0.1096484 | 0.361190 | 7.187429e-01 |\n",
       "| med | 0.63549459 | 0.1005339 | 6.321194 | 7.922642e-09 |\n",
       "\n",
       "\n",
       "$`Mod3: M~X`\n",
       ":   \n",
       "A matrix: 2 × 4 of type dbl\n",
       "\n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(&gt;|t|) |\n",
       "|---|---|---|---|---|\n",
       "| (Intercept) | 1.4995183 | 0.58919773 | 2.545017 | 1.248749e-02 |\n",
       "| pred | 0.5610153 | 0.09448048 | 5.937897 | 4.391362e-08 |\n",
       "\n",
       "\n",
       "$Indirect.Effect\n",
       ":   0.356522196728078\n",
       "$SE\n",
       ":   0.0823778110411936\n",
       "$z.value\n",
       ":   4.32789111803173\n",
       "$N\n",
       ":   100\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$`Mod1: Y~X`\n",
       "             Estimate Std. Error  t value     Pr(>|t|)\n",
       "(Intercept) 2.8572046  0.6932130 4.121684 7.875771e-05\n",
       "pred        0.3961261  0.1111598 3.563574 5.671128e-04\n",
       "\n",
       "$`Mod2: Y~X+M`\n",
       "              Estimate Std. Error  t value     Pr(>|t|)\n",
       "(Intercept) 1.90426882  0.6054585 3.145168 2.203831e-03\n",
       "pred        0.03960392  0.1096484 0.361190 7.187429e-01\n",
       "med         0.63549459  0.1005339 6.321194 7.922642e-09\n",
       "\n",
       "$`Mod3: M~X`\n",
       "             Estimate Std. Error  t value     Pr(>|t|)\n",
       "(Intercept) 1.4995183 0.58919773 2.545017 1.248749e-02\n",
       "pred        0.5610153 0.09448048 5.937897 4.391362e-08\n",
       "\n",
       "$Indirect.Effect\n",
       "[1] 0.3565222\n",
       "\n",
       "$SE\n",
       "[1] 0.08237781\n",
       "\n",
       "$z.value\n",
       "[1] 4.327891\n",
       "\n",
       "$N\n",
       "[1] 100\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 使用函数\n",
    "# library(multilevel)\n",
    "sobel(myData$X, myData$M, myData$Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7190cd84",
   "metadata": {},
   "source": [
    "## 使用mediation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "d8a3ecf4",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Running nonparametric bootstrap\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Causal Mediation Analysis \n",
       "\n",
       "Nonparametric Bootstrap Confidence Intervals with the Percentile Method\n",
       "\n",
       "               Estimate 95% CI Lower 95% CI Upper p-value    \n",
       "ACME             0.3565       0.2142         0.53  <2e-16 ***\n",
       "ADE              0.0396      -0.1984         0.29   0.752    \n",
       "Total Effect     0.3961       0.1602         0.63   0.004 ** \n",
       "Prop. Mediated   0.9000       0.4927         2.11   0.004 ** \n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Sample Size Used: 100 \n",
       "\n",
       "\n",
       "Simulations: 1000 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res.1 = mediate(model.XM, model.MY.XY, treat='X', mediator='M', \n",
    "                boot=TRUE, sims=1000)\n",
    "summary(res.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70beab0d",
   "metadata": {},
   "source": [
    "## 使用lavaan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "6fa64c3f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 1: Specify mediation model\n",
    "# The lavaan syntax\n",
    "model = \"\n",
    "# Path c' (direct effect)\n",
    "Y ~ c*X\n",
    "# Path a\n",
    "M ~ a*X\n",
    "# Path b\n",
    "Y ~ b*M\n",
    "# Indirect effect (a*b)\n",
    "ab := a*b\n",
    "\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "f1f567a6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$header</dt>\n",
       "\t\t<dd><dl>\n",
       "\t<dt>$lavaan.version</dt>\n",
       "\t\t<dd>'0.6.17'</dd>\n",
       "\t<dt>$sam.approach</dt>\n",
       "\t\t<dd>FALSE</dd>\n",
       "\t<dt>$optim.method</dt>\n",
       "\t\t<dd>'nlminb'</dd>\n",
       "\t<dt>$optim.iterations</dt>\n",
       "\t\t<dd>1</dd>\n",
       "\t<dt>$optim.converged</dt>\n",
       "\t\t<dd>TRUE</dd>\n",
       "</dl>\n",
       "</dd>\n",
       "\t<dt>$optim</dt>\n",
       "\t\t<dd><dl>\n",
       "\t<dt>$estimator</dt>\n",
       "\t\t<dd>'ML'</dd>\n",
       "\t<dt>$estimator.args</dt>\n",
       "\t\t<dd><ol>\n",
       "</ol>\n",
       "</dd>\n",
       "\t<dt>$optim.method</dt>\n",
       "\t\t<dd>'nlminb'</dd>\n",
       "\t<dt>$npar</dt>\n",
       "\t\t<dd>5</dd>\n",
       "\t<dt>$eq.constraints</dt>\n",
       "\t\t<dd>FALSE</dd>\n",
       "\t<dt>$nrow.ceq.jac</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$nrow.cin.jac</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$nrow.con.jac</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$con.jac.rank</dt>\n",
       "\t\t<dd>0</dd>\n",
       "</dl>\n",
       "</dd>\n",
       "\t<dt>$data</dt>\n",
       "\t\t<dd><dl>\n",
       "\t<dt>$ngroups</dt>\n",
       "\t\t<dd>1</dd>\n",
       "\t<dt>$nobs</dt>\n",
       "\t\t<dd>100</dd>\n",
       "</dl>\n",
       "</dd>\n",
       "\t<dt>$test</dt>\n",
       "\t\t<dd><strong>$standard</strong> = <dl>\n",
       "\t<dt>$test</dt>\n",
       "\t\t<dd>'standard'</dd>\n",
       "\t<dt>$stat</dt>\n",
       "\t\t<dd>8.88178419700125e-14</dd>\n",
       "\t<dt>$stat.group</dt>\n",
       "\t\t<dd>8.88178419700125e-14</dd>\n",
       "\t<dt>$df</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$refdistr</dt>\n",
       "\t\t<dd>'chisq'</dd>\n",
       "\t<dt>$pvalue</dt>\n",
       "\t\t<dd>&lt;NA&gt;</dd>\n",
       "</dl>\n",
       "</dd>\n",
       "\t<dt>$fit</dt>\n",
       "\t\t<dd><style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>npar</dt><dd>5</dd><dt>fmin</dt><dd>4.44089209850063e-16</dd><dt>chisq</dt><dd>8.88178419700125e-14</dd><dt>df</dt><dd>0</dd><dt>pvalue</dt><dd>&lt;NA&gt;</dd><dt>baseline.chisq</dt><dd>77.4131764309656</dd><dt>baseline.df</dt><dd>3</dd><dt>baseline.pvalue</dt><dd>1.11022302462516e-16</dd><dt>cfi</dt><dd>0.999999999999999</dd><dt>tli</dt><dd>1</dd><dt>logl</dt><dd>-379.612296327384</dd><dt>unrestricted.logl</dt><dd>-379.612296327384</dd><dt>aic</dt><dd>769.224592654767</dd><dt>bic</dt><dd>782.250443584708</dd><dt>ntotal</dt><dd>100</dd><dt>bic2</dt><dd>766.459187569449</dd><dt>rmsea</dt><dd>0</dd><dt>rmsea.ci.lower</dt><dd>0</dd><dt>rmsea.ci.upper</dt><dd>0</dd><dt>rmsea.ci.level</dt><dd>0.9</dd><dt>rmsea.pvalue</dt><dd>&lt;NA&gt;</dd><dt>rmsea.close.h0</dt><dd>0.05</dd><dt>rmsea.notclose.pvalue</dt><dd>&lt;NA&gt;</dd><dt>rmsea.notclose.h0</dt><dd>0.08</dd><dt>srmr</dt><dd>0</dd></dl>\n",
       "</dd>\n",
       "\t<dt>$pe</dt>\n",
       "\t\t<dd><table class=\"dataframe\">\n",
       "<caption>A data.frame: 9 × 9</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>lhs</th><th scope=col>op</th><th scope=col>rhs</th><th scope=col>label</th><th scope=col>exo</th><th scope=col>est</th><th scope=col>se</th><th scope=col>z</th><th scope=col>pvalue</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>Y </td><td>~ </td><td>X  </td><td>c </td><td>0</td><td>0.03960392</td><td>0.10799119</td><td>0.3667329</td><td>7.138183e-01</td></tr>\n",
       "\t<tr><td>M </td><td>~ </td><td>X  </td><td>a </td><td>0</td><td>0.56101532</td><td>0.09353090</td><td>5.9981813</td><td>1.995397e-09</td></tr>\n",
       "\t<tr><td>Y </td><td>~ </td><td>M  </td><td>b </td><td>0</td><td>0.63549459</td><td>0.09901445</td><td>6.4182005</td><td>1.378946e-10</td></tr>\n",
       "\t<tr><td>Y </td><td>~~</td><td>Y  </td><td>  </td><td>0</td><td>2.58142482</td><td>0.36506860</td><td>7.0710678</td><td>1.537437e-12</td></tr>\n",
       "\t<tr><td>M </td><td>~~</td><td>M  </td><td>  </td><td>0</td><td>2.63306954</td><td>0.37237227</td><td>7.0710678</td><td>1.537437e-12</td></tr>\n",
       "\t<tr><td>X </td><td>~~</td><td>X  </td><td>  </td><td>1</td><td>3.00990000</td><td>0.00000000</td><td>       NA</td><td>          NA</td></tr>\n",
       "\t<tr><td>ab</td><td>:=</td><td>a*b</td><td>ab</td><td>0</td><td>0.35652220</td><td>0.08135460</td><td>4.3823238</td><td>1.174201e-05</td></tr>\n",
       "\t<tr><td>Y </td><td>r2</td><td>Y  </td><td>  </td><td>0</td><td>0.37299924</td><td>        NA</td><td>       NA</td><td>          NA</td></tr>\n",
       "\t<tr><td>M </td><td>r2</td><td>M  </td><td>  </td><td>0</td><td>0.26458788</td><td>        NA</td><td>       NA</td><td>          NA</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$header] \\begin{description}\n",
       "\\item[\\$lavaan.version] '0.6.17'\n",
       "\\item[\\$sam.approach] FALSE\n",
       "\\item[\\$optim.method] 'nlminb'\n",
       "\\item[\\$optim.iterations] 1\n",
       "\\item[\\$optim.converged] TRUE\n",
       "\\end{description}\n",
       "\n",
       "\\item[\\$optim] \\begin{description}\n",
       "\\item[\\$estimator] 'ML'\n",
       "\\item[\\$estimator.args] \\begin{enumerate}\n",
       "\\end{enumerate}\n",
       "\n",
       "\\item[\\$optim.method] 'nlminb'\n",
       "\\item[\\$npar] 5\n",
       "\\item[\\$eq.constraints] FALSE\n",
       "\\item[\\$nrow.ceq.jac] 0\n",
       "\\item[\\$nrow.cin.jac] 0\n",
       "\\item[\\$nrow.con.jac] 0\n",
       "\\item[\\$con.jac.rank] 0\n",
       "\\end{description}\n",
       "\n",
       "\\item[\\$data] \\begin{description}\n",
       "\\item[\\$ngroups] 1\n",
       "\\item[\\$nobs] 100\n",
       "\\end{description}\n",
       "\n",
       "\\item[\\$test] \\textbf{\\$standard} = \\begin{description}\n",
       "\\item[\\$test] 'standard'\n",
       "\\item[\\$stat] 8.88178419700125e-14\n",
       "\\item[\\$stat.group] 8.88178419700125e-14\n",
       "\\item[\\$df] 0\n",
       "\\item[\\$refdistr] 'chisq'\n",
       "\\item[\\$pvalue] <NA>\n",
       "\\end{description}\n",
       "\n",
       "\\item[\\$fit] \\begin{description*}\n",
       "\\item[npar] 5\n",
       "\\item[fmin] 4.44089209850063e-16\n",
       "\\item[chisq] 8.88178419700125e-14\n",
       "\\item[df] 0\n",
       "\\item[pvalue] <NA>\n",
       "\\item[baseline.chisq] 77.4131764309656\n",
       "\\item[baseline.df] 3\n",
       "\\item[baseline.pvalue] 1.11022302462516e-16\n",
       "\\item[cfi] 0.999999999999999\n",
       "\\item[tli] 1\n",
       "\\item[logl] -379.612296327384\n",
       "\\item[unrestricted.logl] -379.612296327384\n",
       "\\item[aic] 769.224592654767\n",
       "\\item[bic] 782.250443584708\n",
       "\\item[ntotal] 100\n",
       "\\item[bic2] 766.459187569449\n",
       "\\item[rmsea] 0\n",
       "\\item[rmsea.ci.lower] 0\n",
       "\\item[rmsea.ci.upper] 0\n",
       "\\item[rmsea.ci.level] 0.9\n",
       "\\item[rmsea.pvalue] <NA>\n",
       "\\item[rmsea.close.h0] 0.05\n",
       "\\item[rmsea.notclose.pvalue] <NA>\n",
       "\\item[rmsea.notclose.h0] 0.08\n",
       "\\item[srmr] 0\n",
       "\\end{description*}\n",
       "\n",
       "\\item[\\$pe] A data.frame: 9 × 9\n",
       "\\begin{tabular}{lllllllll}\n",
       " lhs & op & rhs & label & exo & est & se & z & pvalue\\\\\n",
       " <chr> & <chr> & <chr> & <chr> & <int> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t Y  & \\textasciitilde{}  & X   & c  & 0 & 0.03960392 & 0.10799119 & 0.3667329 & 7.138183e-01\\\\\n",
       "\t M  & \\textasciitilde{}  & X   & a  & 0 & 0.56101532 & 0.09353090 & 5.9981813 & 1.995397e-09\\\\\n",
       "\t Y  & \\textasciitilde{}  & M   & b  & 0 & 0.63549459 & 0.09901445 & 6.4182005 & 1.378946e-10\\\\\n",
       "\t Y  & \\textasciitilde{}\\textasciitilde{} & Y   &    & 0 & 2.58142482 & 0.36506860 & 7.0710678 & 1.537437e-12\\\\\n",
       "\t M  & \\textasciitilde{}\\textasciitilde{} & M   &    & 0 & 2.63306954 & 0.37237227 & 7.0710678 & 1.537437e-12\\\\\n",
       "\t X  & \\textasciitilde{}\\textasciitilde{} & X   &    & 1 & 3.00990000 & 0.00000000 &        NA &           NA\\\\\n",
       "\t ab & := & a*b & ab & 0 & 0.35652220 & 0.08135460 & 4.3823238 & 1.174201e-05\\\\\n",
       "\t Y  & r2 & Y   &    & 0 & 0.37299924 &         NA &        NA &           NA\\\\\n",
       "\t M  & r2 & M   &    & 0 & 0.26458788 &         NA &        NA &           NA\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$header\n",
       ":   $lavaan.version\n",
       ":   '0.6.17'\n",
       "$sam.approach\n",
       ":   FALSE\n",
       "$optim.method\n",
       ":   'nlminb'\n",
       "$optim.iterations\n",
       ":   1\n",
       "$optim.converged\n",
       ":   TRUE\n",
       "\n",
       "\n",
       "\n",
       "$optim\n",
       ":   $estimator\n",
       ":   'ML'\n",
       "$estimator.args\n",
       ":   \n",
       "\n",
       "\n",
       "$optim.method\n",
       ":   'nlminb'\n",
       "$npar\n",
       ":   5\n",
       "$eq.constraints\n",
       ":   FALSE\n",
       "$nrow.ceq.jac\n",
       ":   0\n",
       "$nrow.cin.jac\n",
       ":   0\n",
       "$nrow.con.jac\n",
       ":   0\n",
       "$con.jac.rank\n",
       ":   0\n",
       "\n",
       "\n",
       "\n",
       "$data\n",
       ":   $ngroups\n",
       ":   1\n",
       "$nobs\n",
       ":   100\n",
       "\n",
       "\n",
       "\n",
       "$test\n",
       ":   **$standard** = $test\n",
       ":   'standard'\n",
       "$stat\n",
       ":   8.88178419700125e-14\n",
       "$stat.group\n",
       ":   8.88178419700125e-14\n",
       "$df\n",
       ":   0\n",
       "$refdistr\n",
       ":   'chisq'\n",
       "$pvalue\n",
       ":   &lt;NA&gt;\n",
       "\n",
       "\n",
       "\n",
       "$fit\n",
       ":   npar\n",
       ":   5fmin\n",
       ":   4.44089209850063e-16chisq\n",
       ":   8.88178419700125e-14df\n",
       ":   0pvalue\n",
       ":   &lt;NA&gt;baseline.chisq\n",
       ":   77.4131764309656baseline.df\n",
       ":   3baseline.pvalue\n",
       ":   1.11022302462516e-16cfi\n",
       ":   0.999999999999999tli\n",
       ":   1logl\n",
       ":   -379.612296327384unrestricted.logl\n",
       ":   -379.612296327384aic\n",
       ":   769.224592654767bic\n",
       ":   782.250443584708ntotal\n",
       ":   100bic2\n",
       ":   766.459187569449rmsea\n",
       ":   0rmsea.ci.lower\n",
       ":   0rmsea.ci.upper\n",
       ":   0rmsea.ci.level\n",
       ":   0.9rmsea.pvalue\n",
       ":   &lt;NA&gt;rmsea.close.h0\n",
       ":   0.05rmsea.notclose.pvalue\n",
       ":   &lt;NA&gt;rmsea.notclose.h0\n",
       ":   0.08srmr\n",
       ":   0\n",
       "\n",
       "\n",
       "$pe\n",
       ":   \n",
       "A data.frame: 9 × 9\n",
       "\n",
       "| lhs &lt;chr&gt; | op &lt;chr&gt; | rhs &lt;chr&gt; | label &lt;chr&gt; | exo &lt;int&gt; | est &lt;dbl&gt; | se &lt;dbl&gt; | z &lt;dbl&gt; | pvalue &lt;dbl&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|\n",
       "| Y  | ~  | X   | c  | 0 | 0.03960392 | 0.10799119 | 0.3667329 | 7.138183e-01 |\n",
       "| M  | ~  | X   | a  | 0 | 0.56101532 | 0.09353090 | 5.9981813 | 1.995397e-09 |\n",
       "| Y  | ~  | M   | b  | 0 | 0.63549459 | 0.09901445 | 6.4182005 | 1.378946e-10 |\n",
       "| Y  | ~~ | Y   | <!----> | 0 | 2.58142482 | 0.36506860 | 7.0710678 | 1.537437e-12 |\n",
       "| M  | ~~ | M   | <!----> | 0 | 2.63306954 | 0.37237227 | 7.0710678 | 1.537437e-12 |\n",
       "| X  | ~~ | X   | <!----> | 1 | 3.00990000 | 0.00000000 |        NA |           NA |\n",
       "| ab | := | a*b | ab | 0 | 0.35652220 | 0.08135460 | 4.3823238 | 1.174201e-05 |\n",
       "| Y  | r2 | Y   | <!----> | 0 | 0.37299924 |         NA |        NA |           NA |\n",
       "| M  | r2 | M   | <!----> | 0 | 0.26458788 |         NA |        NA |           NA |\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "lavaan 0.6.17 ended normally after 1 iteration\n",
       "\n",
       "  Estimator                                         ML\n",
       "  Optimization method                           NLMINB\n",
       "  Number of model parameters                         5\n",
       "\n",
       "  Number of observations                           100\n",
       "\n",
       "Model Test User Model:\n",
       "                                                      \n",
       "  Test statistic                                 0.000\n",
       "  Degrees of freedom                                 0\n",
       "\n",
       "Model Test Baseline Model:\n",
       "\n",
       "  Test statistic                                77.413\n",
       "  Degrees of freedom                                 3\n",
       "  P-value                                        0.000\n",
       "\n",
       "User Model versus Baseline Model:\n",
       "\n",
       "  Comparative Fit Index (CFI)                    1.000\n",
       "  Tucker-Lewis Index (TLI)                       1.000\n",
       "\n",
       "Loglikelihood and Information Criteria:\n",
       "\n",
       "  Loglikelihood user model (H0)               -379.612\n",
       "  Loglikelihood unrestricted model (H1)       -379.612\n",
       "                                                      \n",
       "  Akaike (AIC)                                 769.225\n",
       "  Bayesian (BIC)                               782.250\n",
       "  Sample-size adjusted Bayesian (SABIC)        766.459\n",
       "\n",
       "Root Mean Square Error of Approximation:\n",
       "\n",
       "  RMSEA                                          0.000\n",
       "  90 Percent confidence interval - lower         0.000\n",
       "  90 Percent confidence interval - upper         0.000\n",
       "  P-value H_0: RMSEA <= 0.050                       NA\n",
       "  P-value H_0: RMSEA >= 0.080                       NA\n",
       "\n",
       "Standardized Root Mean Square Residual:\n",
       "\n",
       "  SRMR                                           0.000\n",
       "\n",
       "Parameter Estimates:\n",
       "\n",
       "  Standard errors                             Standard\n",
       "  Information                                 Expected\n",
       "  Information saturated (h1) model          Structured\n",
       "\n",
       "Regressions:\n",
       "                   Estimate  Std.Err  z-value  P(>|z|)\n",
       "  Y ~                                                 \n",
       "    X          (c)    0.040    0.108    0.367    0.714\n",
       "  M ~                                                 \n",
       "    X          (a)    0.561    0.094    5.998    0.000\n",
       "  Y ~                                                 \n",
       "    M          (b)    0.635    0.099    6.418    0.000\n",
       "\n",
       "Variances:\n",
       "                   Estimate  Std.Err  z-value  P(>|z|)\n",
       "   .Y                 2.581    0.365    7.071    0.000\n",
       "   .M                 2.633    0.372    7.071    0.000\n",
       "\n",
       "R-Square:\n",
       "                   Estimate\n",
       "    Y                 0.373\n",
       "    M                 0.265\n",
       "\n",
       "Defined Parameters:\n",
       "                   Estimate  Std.Err  z-value  P(>|z|)\n",
       "    ab                0.357    0.081    4.382    0.000\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Step 2: Estimate model - delta method (by default)\n",
    "fitmod = sem(model, data=myData)\n",
    "# Step 3: Request summary\n",
    "summary(fitmod, fit.measures=TRUE, rsquare=TRUE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "50b59da3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A lavaan.data.frame: 7 × 10</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>lhs</th><th scope=col>op</th><th scope=col>rhs</th><th scope=col>label</th><th scope=col>est</th><th scope=col>se</th><th scope=col>z</th><th scope=col>pvalue</th><th scope=col>ci.lower</th><th scope=col>ci.upper</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>Y </td><td>~ </td><td>X  </td><td>c </td><td>0.03960392</td><td>0.12755884</td><td>0.3104757</td><td>7.561992e-01</td><td>-0.2000781</td><td>0.2901507</td></tr>\n",
       "\t<tr><td>M </td><td>~ </td><td>X  </td><td>a </td><td>0.56101532</td><td>0.09658860</td><td>5.8082973</td><td>6.311139e-09</td><td> 0.3874452</td><td>0.7691867</td></tr>\n",
       "\t<tr><td>Y </td><td>~ </td><td>M  </td><td>b </td><td>0.63549459</td><td>0.10507922</td><td>6.0477665</td><td>1.468676e-09</td><td> 0.4311823</td><td>0.8426629</td></tr>\n",
       "\t<tr><td>Y </td><td>~~</td><td>Y  </td><td>  </td><td>2.58142482</td><td>0.32884071</td><td>7.8500767</td><td>4.218847e-15</td><td> 1.9048312</td><td>3.1598721</td></tr>\n",
       "\t<tr><td>M </td><td>~~</td><td>M  </td><td>  </td><td>2.63306954</td><td>0.34658373</td><td>7.5972104</td><td>3.019807e-14</td><td> 1.9150087</td><td>3.3124284</td></tr>\n",
       "\t<tr><td>X </td><td>~~</td><td>X  </td><td>  </td><td>3.00990000</td><td>0.00000000</td><td>       NA</td><td>          NA</td><td> 3.0099000</td><td>3.0099000</td></tr>\n",
       "\t<tr><td>ab</td><td>:=</td><td>a*b</td><td>ab</td><td>0.35652220</td><td>0.08216977</td><td>4.3388485</td><td>1.432312e-05</td><td> 0.2168658</td><td>0.5321631</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A lavaan.data.frame: 7 × 10\n",
       "\\begin{tabular}{llllllllll}\n",
       " lhs & op & rhs & label & est & se & z & pvalue & ci.lower & ci.upper\\\\\n",
       " <chr> & <chr> & <chr> & <chr> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t Y  & \\textasciitilde{}  & X   & c  & 0.03960392 & 0.12755884 & 0.3104757 & 7.561992e-01 & -0.2000781 & 0.2901507\\\\\n",
       "\t M  & \\textasciitilde{}  & X   & a  & 0.56101532 & 0.09658860 & 5.8082973 & 6.311139e-09 &  0.3874452 & 0.7691867\\\\\n",
       "\t Y  & \\textasciitilde{}  & M   & b  & 0.63549459 & 0.10507922 & 6.0477665 & 1.468676e-09 &  0.4311823 & 0.8426629\\\\\n",
       "\t Y  & \\textasciitilde{}\\textasciitilde{} & Y   &    & 2.58142482 & 0.32884071 & 7.8500767 & 4.218847e-15 &  1.9048312 & 3.1598721\\\\\n",
       "\t M  & \\textasciitilde{}\\textasciitilde{} & M   &    & 2.63306954 & 0.34658373 & 7.5972104 & 3.019807e-14 &  1.9150087 & 3.3124284\\\\\n",
       "\t X  & \\textasciitilde{}\\textasciitilde{} & X   &    & 3.00990000 & 0.00000000 &        NA &           NA &  3.0099000 & 3.0099000\\\\\n",
       "\t ab & := & a*b & ab & 0.35652220 & 0.08216977 & 4.3388485 & 1.432312e-05 &  0.2168658 & 0.5321631\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A lavaan.data.frame: 7 × 10\n",
       "\n",
       "| lhs &lt;chr&gt; | op &lt;chr&gt; | rhs &lt;chr&gt; | label &lt;chr&gt; | est &lt;dbl&gt; | se &lt;dbl&gt; | z &lt;dbl&gt; | pvalue &lt;dbl&gt; | ci.lower &lt;dbl&gt; | ci.upper &lt;dbl&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|\n",
       "| Y  | ~  | X   | c  | 0.03960392 | 0.12755884 | 0.3104757 | 7.561992e-01 | -0.2000781 | 0.2901507 |\n",
       "| M  | ~  | X   | a  | 0.56101532 | 0.09658860 | 5.8082973 | 6.311139e-09 |  0.3874452 | 0.7691867 |\n",
       "| Y  | ~  | M   | b  | 0.63549459 | 0.10507922 | 6.0477665 | 1.468676e-09 |  0.4311823 | 0.8426629 |\n",
       "| Y  | ~~ | Y   | <!----> | 2.58142482 | 0.32884071 | 7.8500767 | 4.218847e-15 |  1.9048312 | 3.1598721 |\n",
       "| M  | ~~ | M   | <!----> | 2.63306954 | 0.34658373 | 7.5972104 | 3.019807e-14 |  1.9150087 | 3.3124284 |\n",
       "| X  | ~~ | X   | <!----> | 3.00990000 | 0.00000000 |        NA |           NA |  3.0099000 | 3.0099000 |\n",
       "| ab | := | a*b | ab | 0.35652220 | 0.08216977 | 4.3388485 | 1.432312e-05 |  0.2168658 | 0.5321631 |\n",
       "\n"
      ],
      "text/plain": [
       "  lhs op rhs label est        se         z         pvalue       ci.lower  \n",
       "1 Y   ~  X   c     0.03960392 0.12755884 0.3104757 7.561992e-01 -0.2000781\n",
       "2 M   ~  X   a     0.56101532 0.09658860 5.8082973 6.311139e-09  0.3874452\n",
       "3 Y   ~  M   b     0.63549459 0.10507922 6.0477665 1.468676e-09  0.4311823\n",
       "4 Y   ~~ Y         2.58142482 0.32884071 7.8500767 4.218847e-15  1.9048312\n",
       "5 M   ~~ M         2.63306954 0.34658373 7.5972104 3.019807e-14  1.9150087\n",
       "6 X   ~~ X         3.00990000 0.00000000        NA           NA  3.0099000\n",
       "7 ab  := a*b ab    0.35652220 0.08216977 4.3388485 1.432312e-05  0.2168658\n",
       "  ci.upper \n",
       "1 0.2901507\n",
       "2 0.7691867\n",
       "3 0.8426629\n",
       "4 3.1598721\n",
       "5 3.3124284\n",
       "6 3.0099000\n",
       "7 0.5321631"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Step 2: Estimate model - percentile bootstrapping method\n",
    "fitmod2 = sem(model, data=myData, se=\"bootstrap\", bootstrap=1000)\n",
    "# Ste[pRequest percentile bootstrap 95% confidence intervals\n",
    "parameterEstimates(fitmod2, ci=TRUE, level=0.95, boot.ci.type=\"perc\")"
   ]
  }
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